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Chapter 11
Elastic Relaxation and Response to Deformation of Soft Gels Mehdi Bouzid1 and Emanuela Del Gado*,2 1LPTMS
- Université Paris Sud - Bâtiment 100 15 rue Georges Clémenceau 91405 Orsay Cedex, France 2Department of Physics and Institute for Soft Matter Synthesis and Metrology, Georgetown University, 37th and O Streets NW, Washington DC 20057, United States *E-mail:
[email protected].
Soft amorphous solids such as gels made of particles or small aggregates form in a variety of soft matter systems. The complexity of the large scale organization of the gel network, its elasticity, and the mechanical heterogeneities that can be easily generated during the gel self-assembly have implications for the aging properties of those materials and their strongly nonlinear mechanical response. Here we review our recent work where we used 3D numerical simulations of a microscopic model to investigate how the competition between thermal fluctuations and elastic relaxation can qualitatively change the gel aging and how the network topology fundamentally determines the nonlinear response under shear. The insight gained through the numerical simulations helps us rationalize various experimental observations and identify new routes for material design.
Introduction Self-assembly and aggregation of soft condensed matter (proteins, colloids or polymers) into poorly connected and weakly elastic solids is very common and ubiquitous in nature (1–3). Phase separation, spinodal decomposition as well as externally driven self-assembly or aggregation often lead to gels, which display diverse structural and mechanical features (4–8).
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In most cases, the interaction energies and the size of the aggregating units make these structures quite sensitive to thermal fluctuations, with a rich relaxation dynamics associated to spontaneous and thermally activated processes. In addition to affecting the time evolution, or aging, of the material properties at rest, those dynamical processes interplay with an imposed mechanical load or deformation and may be hence crucial for the mechanical response of this class of solids (5, 9–15). Under deformation, soft gels typically display strongly nonlinear response and their structural complexity allows for a wide variety of different behaviors, ranging from softening to stiffening, ductility and brittleness, all of which have important implications for novel technological applications and smart material design (16, 17). Since a deeper understanding of the structure-mechanics relationship is needed to control material properties and achieve their design, microscopic numerical simulations that do not assume a specific constitutive model but provide space and time resolved insight into dynamical processes from which the overall constitutive behavior emerges have become a major investigation tool, especially in combination with experiments (18). Having recently developed a computational approach that has proven very effective in investigating the mechanics and rheology of complex network structures (which can or cannot restructure under deformation), here we would like to provide an overview of the insight gained, in particular with respect to the aging dynamics of soft gels and to the emergence of the strongly nonlinear response from the gel microstructure.
Computational Approach: Model and Numerical Simulations The idea is to obtain the gel structure via self-assembly of elementary units (particles, small aggregates, fibers…) through effective interactions that contain minimal, but crucial, information of the physical-chemistry of the material of interest. Starting from the self-assembled structures, we have used Molecular Dynamics (MD) and Non-equilibrium Molecular Dynamics (NEMD) type of simulations to analyze the structure and mechanics of large samples at rest or under deformations, and with different types of boundary conditions. Such approach allows for novel, more detailed insight into how the microstructure and the effective interactions among the structural units determine their mechanics and for a full description of the non-linear response (12, 19–22). We have developed a minimal 3D model for colloidal particles gels, considering particles (or small aggregates represented as particles) of diameter a that interact with a short range attraction (combined with a repulsive core) U2(rij), where rij is the distance between particles i and j. In particular for computational efficiency, for U2(rij) we use the form U2(rij) = A((a/r)18 − (1/r)16), where r is the distance rescaled by the particle diameter d, while a and A are dimensionless parameters that control the width and the depth of the potential respectively (19). Once particles start to aggregate, their mechanical contacts can be more complex than the simple geometric contact between two perfect spheres and, in particular, there may be an energy cost associated to rotation of particles 212
around each other in an aggregate made by more than two of them, due to surface interactions more complex than simply described by the short range, spherically symmetric attractive well U2(rij) (23, 24). We include therefore an additional short range repulsion U3(rij, rik, θijk), which depends on the angle between bonds and departing from the same particle i and introduce an angular rigidity to the particle connections. For U3(rij, rik, θijk), we use the computationally convenient form , where B,θ and u are dimensionless parameters that set the range of the angular rigidity and its relative strength with respect to the depth of the attractive well. The radial ensures that U3 vanishes modulation being the Heaviside function) (19). beyond a distance 2d (with The total potential energy of a system of N particles with coordinates is and then computed as used in simulations where we solve the equations of motion for all particles in presence of thermal fluctuations using Langevin dynamics. This potential energy depends parametrically on the five dimensionless quantities mentioned above, which are fixed to the following values: A = 6.27, a = 0.85, B = 67.27, θ = 65° and u = 0.3. Tuning these parameters leads to a variety of mechanically stable porous microstructures. The values chosen here are such that a disordered and thin percolating network starts to self-assemble for low particle volume fractions at ε ≈ 20kBT (19). The diameter d of the particles is determined by the distance at which two particles start to experience the short range repulsion and it is the unit lengthscale used in the simulations, while ε is the unit energy. For computational convenience we solve the Langevin equations of motion with inertia for each particle, with each particle having a unit mass m, but then fix the drag coefficient to obtain an overdamped dynamics, with the time scale associated to oscillations due to inertia and used as time unit. More details about the effective interactions, the parameters and protocols used in the simulations can be found in (12, 19–22, 25). In the simulations, the particle volume fraction φ is estimated as the fraction of the total volume V that is occupied by the N particles φ = πa3/6V and in the following we will mainy discuss (V is the volume of the simulation box). As just mentioned above, the particle spontaneously self-assemble into a disordered, persistent and interconnected network. In the remainder we will mainly refer to simulations performed using particles. In spite of its simplicity, our model gel captures several physical features of real soft gels: the heterogeneity of the gel structure is associated to mechanical inhomogeneities, and the simulations have revealed the coexistence of stiffer regions (where tensile stresses tend to accumulate) with softer domains, where, in presence of thermal fluctuations, most of the structural relaxation occurs (19, 20). In this type of simulations we can solve the many-body dynamics and compute the stresses due to the interparticle interactions through the network, with and without an imposed deformation, using the virial stress tensor in which α, β stand for the cartesian components {x, y, z} (26, 27). Hence we 213
can distinguish the contribution of particles located in different part of the gel structure (e.g., in the network strands or in the branching points) to the stresses. If we restrict the sum to a relatively small portion of the total volume we can compute the local stresses and their spatial distribution. Figure 1 shows a snapshot of a typical gel structure in the model just described, where the network has been colored to distinguish prevalently tensile or compressive contributions to local stresses.
Figure 1. Schematic of the model gel network at rest and at low volume fraction φ = 7%. The structure is represented by showing the inter particle bonds, each bond is represented by a segment and generated when the distance between two particles particles i and j is less than dij ≤ 1.3a, the color indicates the value of the contribution (per particle) to the component σxx of the stress tensor. (see color insert)
Aging and Elastic Relaxation Due to the complexity of gels microstructure and to the structural disorder always present, diffusive processes driven by thermal energy are expected, in general, to lead to slow cooperative motion with a subdiffusive microscopic dynamics, governed by a wide distribution of relaxation times and cooperative in nature. Such phenomena manifest themselves in the slow decay of time correlations in density fluctuations, displacements or structural rearrangements, typically measured in quasi-elastic scattering experiments or in numerical simulations (e.g., molecular or Brownian dynamics), typically slower than with β exponential and described by a stretched exponential decay < 1, akin to the slow dynamics close to a glass transition (28–31). Using the 214
computational approach described above we were able to show that in soft gels the consequences of local bond breaking propagate along the gel network over distances larger than the average mesh size and this long-range correlations are at the origin of the cooperative dynamics reported in the experiments (19, 20). Interstingly enough, time- and space-resolved scattering measurements have also detected, in gels and other soft materials, dynamics faster than exponential (so called compressed exponential dynamics), intermittency and abrupt microstructural changes. Such experiments have raised the question whether the aging may be controlled, instead, by stress relaxation through elastic rebound of parts of the material, after local breakages occur in its structure, challenging the generally accepted paradigm that the relaxation dynamics is always slow and glassy (i.e. similar to microscopic dynamics in supercooled liquids) in these soft materials (4, 32–40). While a mean field theoretical framework invoking elasticity has been proposed in (38) and successfully used to explain part of the experimental observations, a 3 dimensional, microscopic understanding of the intermittent dynamics and of its connection to faster than exponential relaxation of density fluctuations was still fundamentally lacking. We have combined the MD based computational approach described above with a MonteCarlo dynamics in which we periodically scan the local stresses in the gels and remove those connections whose contribution to the overall tension in the material is the largest. This process mimics the gel aging through micro-collapses in its structure and allowed us to analyse the dynamical processes that develop from the local ruptures and help redistribute the stresses over time. Figure 2 shows a cartoon of the aging process in our simulations. We note that our study does not necessarily cover all possible elementary aging events that can happen in different materials, e.g., we do not consider coarsening or compaction of initially loosely packed domains (41). As a matter of fact, in our simulations, recombinations of the gel branches are actually possible but just not observed at the volume fractions and for the time window explored here. One could extend our simple but effective approach to include also local micro-compaction events that would tend to favor an increase of the modulus over time, whereas the breaking events – micro-collapses considered here obviously weaken the gel over time. In spite of the different trend in the time evolution of the gel strength, the disruption of the elastic strain field due to the rupture of a branch of the gel is basically the same as the one induced by a recombination of the gel branches (42). Hence we do not expect qualitatively different results in the two cases. In our study, we were able to monitor the consequences of the aging process acting on the precise same gel structure in presence of different amount of thermal fluctuations, by solving the Langevin equations of motion for all particles in the gel for different values of kBT/ε as described in (21). The value of kBT/ε represents different amount of thermal fluctuations (i.e. Brownian stresses, whose order of magnitude is ~kBT) with respect to the enthalpic stresses (whose order of magnitude is ~ε) initially frozen-in in the network structure during the gel formation and processing. kBT/ε can be easily fixed as a tuning parameters in our simulations. Figure 3 shows the data obtained for the time-decay of the correlation of the density fluctuations (the coherent scattering function) at different wave vectors q and of the correlations of stress fluctuations for different values of kBT/ε. 215
Figure 2. Snapshots of the colloidal gel network showing the interparticle bonds before (left) and after (right) a bond rupture. On the left, the bond that will break is highlighted. On the right, the arrows indicate the displacement after the rupture. Reproduced with permission from Ref. (21). Creative Commons license available at https://creativecommons.org/licenses/by/4.0/. Copyright 2017 Nature Publishing Group.
Our key new findings are that the relaxation dynamics underlying the aging change dramatically if the enthalpic stress heterogeneities, frozen-in upon solidification, are significantly larger than Brownian stresses. The decay of the coherent (and incoherent) scattering functions over time change from stretched to compressed exponential as kBT/ε decreases (see Fig. 3, left). From the behaviour obtained at different wave vectors we can elucidate that, upon decreasing the strength of thermal fluctuations, the dynamics gradually changes from stretched to compressed exponential relaxation. The wave vector dependence well agrees with experimental observations: the exponent β decreases with increasing q and the relaxation time increases with decreasing q but becomes less sensitive to changes in q. By analysing the distribution of microscopic displacements following the micro-collapses, we show that the microscopic motion underlying the compressed exponential dynamics is of elastic origin since the PDF is fully captured by the prediction of continuum elasticity for the elastic strain (42). We elaborate that the timescales governing stress relaxation, respectively through thermal fluctuations and elastic recovery, are key. When thermal fluctuations are weak with respect to enthalpic stress heterogeneities, the stress can partially relax through elastically driven fluctuations. Such fluctuations are intermittent, because of strong spatio- temporal correlations that persist well beyond the timescale of experiments or simulations, and the elasticity built into the solid structure controls microscopic displacements, leading to the faster than exponential dynamics reported in experiments and indeed hypothesized by the theory (4, 32, 33, 37–40, 43). Thermal fluctuations, instead, disrupt the spatial distributions of local stresses and their persistence in time, favoring a gradual loss of correlations and a slow evolution of the material properties. The difference in the stress relaxation measured in the simulations with different amount of thermal fluctuations, while the same amount of connections have been removed, 216
and corresponding to stretched vs compressed exponential relaxation is shown in Figure 3 (right). In the simulations, we have been able to clearly demonstrate how the elastic nature of the partial stress relaxation affects the microscopic particle dynamics, when thermal fluctuations are too weak.
Figure 3. Stretched and compressed exponential dynamics in soft gels. Left: Time decay of the correlations of the density fluctuations as provided by the coherent scattering functions measured in the simulations for different wave vectors q and for different values of kBT/ε starting from exactly the same gel configuration. The decay changes from stretched to compressed exponential upon reducing the Brownian stresses with respect to the enthalpic stresses initially frozen-in in the gel structures. Right: Time correlations computed from the fluctuations of a representative component of the total stress in the gel during aging. Decreasing kBT/ε leads to strongly correlated, intermittent fluctuations, strongly reminiscent of the experimental findings. The time correlations show that the stress fluctuations are persistently correlated when only elastic relaxation of the network is at play. Modified with permission from Ref. (21). Creative Commons license available at https://creativecommons.org/licenses/by/4.0/ Copyright 2017 Nature Publishing Group. (see color insert) The study just described provided deeper understanding of the interplay between thermal fluctuations and elasticity in the aging of soft materials, and in particular of how stress relaxation, when driven essentially by elasticity, translates into qualitatively different microscopic dynamics due to the radically different nature of the stress spatio-temporal correlations. When elastic relaxation dominates, the stress fluctuations are strongly and persistently correlated over large distances, and this fact explains why experimental observations are often close to the prediction of the mean field theory, even if the materials are structurally heterogeneous (21). The insight gained here helps us identify the conditions for which the elastically driven intermittent dynamics emerge in different jammed solids. Close to ergodicity, enthalpic and thermal degrees of freedom may still couple and stress correlations decay relatively fast. When the material is deeply quenched and jammed, instead, recovering the coupling between the distinct degrees of freedom and restoring equilibrium through 217
thermal fluctuations requires timescales well beyond the ones accessible in typical experiments or simulations. If elastic degrees of freedom are the ones responsible for the stress relaxation, the associated stress fluctuations are strongly spatio-temporally correlated and the result will be intermittent dynamics and compressed exponential relaxations (33, 40).
Network Topology and Nonlinear Response So far we have analyzed the relaxation dynamics and the stress redistribution in soft gels at rest. Such dynamics are controlled by the nature of the microscopic fluctuations (spontaneous or driven by internal stress relaxation) for a fixed configuration. More generally, the stress redistribution is also strongly affected by, and can vary significantly with, the structural complexity and diversity. It can be very dramatically seen in presence of an external load, when the microstructural diversity leads to a rich variety of strongly nonlinear responses (10, 44–48). Being able to design the microstructure-process or microstructure-rheology interplay in this class of materials is essential to achieve smart rheology and mechanics that can be finely tuned and adjusted on the fly, such as in soft inks for 3D printing technologies (16, 17). Within this context in particular, being able to fine tune the interplay and feedback among deformation, deformation rate and microscopic restructuring of soft gels requires a deeper understanding of the role of the gel microstructure in its mechanical response. Our computational approach has the potential to provide unique insight thanks to the possibility to access the spatio-temporal microscopic dynamics and its link to the emerging rheological response of the material. We have used the model previously introduced to explore specifically the role of the topology of the gel networks in their linear and non-linear response. For these simulations, we have studied gels at different volume fractions and in athermal conditions, subjected to different type of mechanical tests (12, 22, 25). In the following, we highlight some of the outcomes of studies where we have imposed a deformation rate on the initially solid sample as in a start-up shear experiments by performing small deformation steps where a small affine shear strain step δγ is imposed on all particles. The affine strain step is followed by a relaxation step of duration δt during which the stresses accumulated under strain are relaxed by solving damped equation of motion, in which the damping coefficient represents the dissipation of the particle motion through the solvent ). We use Lees-Edwards (for the data discussed in the following boundary conditions that are compatible with the flow (49). Varying δt allows us to vary the shear rate to investigate the shear rate dependence (12), but here we will focus on mechanical tests performed at a fixed shear . By repeating the deformation and relaxation steps we can rate accumulate arbitrarily large shear strains γ and, at each step, compute local and global stresses as discussed above. To characterize the linear response of the gels we have also performed small amplitude oscillatory shear tests (described in (12) and (22)), to reconstruct the viscoelastic spectra and extract the low frequency storage modulus G0. 218
In the gel model briefly described above, varying the volume fraction between 5% and 15% changes the number of branching points in the gel network without changing the morphology of the branches. The self-assembly protocol and the effective interactions used are the same. Therefore, varying the volume fraction allows us to explore the role of one specific topological change in the structure (i.e., a change in the number of branching points) for the mechanics. The gels at 5% in particle volume fractions are very sparse, being composed of relatively long and semiflexible strands connected by a few branching points. Upon increasing the volume fraction, the strands become shorter (and hence less flexible) as more and more branching points form and distribute more homogenously in space (50).
Figure 4. Linear and non-linear response to a start-up shear test in simulations of soft gels. Load curve identifying the different regime of the gel response at intermediate volume fractions, where the initial linear regime is followed by stiffening and hardening. Figure 4 shows the typical load curve obtained in the shear start-up tests: a relatively narrow linear regime is followed by an extended regime where the dependence of the shear stress σ on the strain γ is nonlinear (22). The first part of the nonlinear regime can be associated to a stiffening of the gels, due to the straightening of large part of the structure, a typical nonlinear but elastic process (51). The nonlinear increase of the stress with the strain in denser and more densely connected gels tend instead to be due to hardening, a plastic process, because of the formation of additional branching points under shear (the hardening is absent in the very dilute gels and becomes more prominent upon increasing the volume fraction). From the load curve we can compute the differential moduus and use it to quantitatively characterize the nonlinear response. In Figure 5 we have plotted K = G0 as a function of the shear stress σ normalized by 219
the value of the stress σc at which the nonlinear response sets in, for a dilute gel at 5% in volume fraction. The stiffening in this case is preceded by a prominent softening of the material and, analyzing the microscopic deformation in the gel structure, we have been able to rationalize such phenomenon in terms of the bent strands produced during the gel self-assembly, when the gel is very sparse. In view of the semiflexible nature of the gel strands, their bending has an energy costs which favours its release under deformation. The following and clearly distinct nonlinear regime features a pronounced increase of the differential modulus that in these very soft gels is mainly ascribed to the stiffer response of large part of the structure that have been already straightened out, as proved by the degree of alignment of the microstructure in the direction of maximum elongation (12, 13, 22). The dependence of the differential modulus on the stress in the stiffening regime reported in Figure 5 is consistent with the stiffening predicted by theories for random networks of semiflexible filaments and typically observed experimentally in biopolymer networks (3, 51–54).
Figure 5. The response of a very soft gel (volume fraction 5%), displayed in terms of the differential shear modulus, normalized by the low frequency storage modulus, as a function of the stress. Such gels feature a pronounced softening followed by stiffening. By increasing the volume fraction and changing the gel topology towards more connected and spatially homogeneous networks, the tendency to softening progressively disappears and the stiffening is characterized by a different scaling of the differential modulus with the stress, akin to what typically predicted for networks of stiff rods. Increasing the volume fraction even further, the gels eventually have microstructure quite similar to cellular solids and their 220
mechanical response is hardly nonlinear, while they feature a less ductile damage accumulation, rapidly leading to a brittle failure. The results obtained prove that the network topology in soft gels plays a crucial role in how the stress can redistribute under deformation and hence in the emergence of the nonlinear rheological response of the material (55). Such findings suggest that topological control on gel microstructure could open interesting possibilities for smart material design that do not require changing the chemistry of the compounds.
Conclusions Three-dimensional microscopic simulations of soft gel networks provide unique access to the spatio-temporal dynamic underlying aging and non-linear rheology of these versatile materials. We have developed a computational approach based on a particle model and here have given an overview of the typeof insight that can be obtained. Our recent studies allowed us to disentangle the interplay between thermal fluctuations and elastic relaxation during the aging of soft gels and other amorphous solids, shedding new light into the compressed exponential, intermittent microscopic dynamics detected in experiments. We have shown that the nature of the fluctuations controls the type of aging dynamics for a given microstructure. Nevertheless, the microstructural complexity of these materials also plays an important role on stress redistribution, and our three-dimensional microscopic simulations have allowed us to elucidate this aspect as well. In particular, we have shown here how the network topology alone can radically change the stress-redistribution, and hence the nature of the non-linear response, in start-up shear experiments. On the basis of the results obtained, controlling the network topology could in principle allow for tuning the non-linear rheology from softening to stiffening and hardening to brittle rupture, without changing the gel components.
Acknowledgments The authors thank the Impact Program of the Georgetown Environmental Initiative and Georgetown University for funding and the Kavli Institute for Theoretical Physics at the University of California Santa Barbara for hospitality. This research was supported in part by the National Science Foundation under Grant No. NSF PHY17-48958.
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